scholarly journals The Law of Large Numbers and the Central Limit Theorem in Banach Spaces

1976 ◽  
Vol 4 (4) ◽  
pp. 587-599 ◽  
Author(s):  
J. Hoffmann-Jorgensen ◽  
G. Pisier
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Banach Spaces ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
The Law

The Central Limit Theorem and the Law of Large Numbers for Pair-Connectivity in Bernoulli Trees

1989 ◽  
Vol 3 (4) ◽  
pp. 477-491
Author(s):  
Kyle T. Siegrist ◽  
Ashok T. Amin ◽  
Peter J. Slater
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Network Reliability ◽  
Random Variable ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
The Mean ◽  
The Law

Consider the standard network reliability model in which each edge of a given (n, m)-graph G is deleted, independently of all others, with probability q = 1– p (0 <p < 1). The pair-connectivity random variable X is defined to be the number of connected pairs of vertices that remain in G. The mean of X has been proposed as a measure of reliability for failure-prone communications networks in which the edge deletions correspond to failures of the communications links. We consider deviations from the mean, the law of large numbers, and the central limit theorem for X as n → ∞. Some explicit results are obtained when G is a tree using martingale difference sequences. Stars and paths are treated in detail.

CLTs for Dependent Processes

2021 ◽  
pp. 548-567
Author(s):  
James Davidson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Strong Mixing ◽  
Mixing Processes ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
The Law ◽  
Dependent Processes

This chapter deals with the central limit theorem (CLT) for dependent processes. As with the law of large numbers, the focus is on near‐epoch dependent and mixing processes and array versions of the results are given to allow heterogeneity. The cornerstone of these results is a general CLT due to McLeish, from which a result for martingales is obtained directly. A result for stationary ergodic mixingales is given, and the rest of the chapter is devoted to proving and interpreting a CLT for mixingales and hence for arrays that are near‐epoch dependent on a strong‐mixing and uniform-mixing processes.

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